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Bull. Aust. Math. Soc. 91 (2015), 303–310 doi:10.1017/S0004972714000926 FRAGMENTABILITY BY THE DISCRETE METRIC WARREN B. MOORS (Received 21 August 2014; accepted 6 November 2014; first published online 5 January 2015) Abstract In a recent paper, topological spaces (X, τ) that are fragmented by a metric that generates the discrete topology were investigated. In the present paper we shall continue this investigation. In particular, we will show, among other things, that such spaces are σ-scattered, that is, a countable union of scattered spaces, and characterise the continuous images of separable metrisable spaces by their fragmentability properties. 2010 Mathematics subject classification: primary 46B20; secondary 46B22. Keywords and phrases: fragmentable, sigma-scattered, topological game. In [7], topological spaces (X, τ) that are fragmented by a metric that generates the discrete topology were investigated. In this paper we show, among other things, that such spaces are σ-scattered. The reason behind the interest in fragmentability lies in the fact that fragmentability (σ-fragmentability) has had numerous applications to many parts of analysis; see [3–6, 8, 17, 23–28, 30, 33, 35–39], to mention but a small selection of them. Let (X, τ) be a topological space and let ρ be a metric defined on X. Following [12], we shall say that (X, τ) is fragmented by ρ if whenever ε > 0 and A is a nonempty subset of X there is a τ-open set U such that U ∩ A , ∅ and ρ − diam(U ∩ A) < ε. A significant generalisation of fragmentability is the following: a topological space (X, τ), endowed with a metric ρ, is σ-fragmented by ρ if, for each ε > 0, there S exists a cover {Xnε : n ∈ N} of X (that is, n∈N Xnε = X) such that for every n ∈ N and every nonempty subset A of Xnε there exists a τ-open set U such that U ∩ A , ∅ and ρ − diam(U ∩ A) < ε; see [9–11]. Theorem 1. Let (X, τ) be a Hausdorff regular space. Then the following are equivalent: (i) (X, τ) is fragmented by a metric that generates the discrete topology; (ii) (X, τ) is σ-fragmented by the discrete metric; (iii) (X, τ) is σ-scattered, that is, a countable union of scattered spaces. Proof. The proof that (i) ⇒ (ii) follows from [21, Proposition 3.1]. To see that (ii) ⇒ (iii), we simply apply the definition of σ-fragmentability with c 2015 Australian Mathematical Publishing Association Inc. 0004-9727/2015 $16.00 303 304 W. B. Moors [2] ε := 1/2 < 1. The fact that (iii) ⇒ (ii) is obvious. Finally, (ii) ⇒ (i) follows from [21, Proposition 3.2]. Thus, the study of fragmentability by a metric that generates the discrete topology reduces to the (well studied) study of scattered spaces. In the presence of Lindelöfness, fragmentability by a metric that generates the discrete topology imposes a severe constraint on the size of the underlying set. Corollary 2. Let (X, τ) be a hereditarily Lindelöf Hausdorff regular space. Then (X, τ) is countable provided that (X, τ) is fragmented by a metric that generates the discrete topology. In particular, every subset of a separable metric space that is fragmented by a metric that generates the discrete topology is countable. Proof. By Theorem 1, we know that X is a countable union of scattered spaces. Hence, it is sufficient to show that a hereditarily Lindelöf scattered space is countable. Let [ U := {U ∈ τ : U is countable} and let U ∗ := U. U∈U Since X is hereditarily Lindelöf, it follows that U ∗ ∈ U. We claim that X = U ∗ . Indeed, if this were not the case, then X\U ∗ , ∅ and so there would exist an open set W such that (X \ U ∗ ) ∩ W is a singleton. Clearly, then, U ∗ ∪ W ∈ U. However, this is impossible since U ∗ ∪ W * U ∗ . At the price of having to introduce several new definitions and several basic results, we can extend Corollary 2 as follows. Let (X, τ) be a topological space. Then we call P ⊆ 2X \ {∅} a partial exhaustive partition of X if: S (i) P∈P P ∈ τ; (ii) the members of P are pairwise disjoint; S (iii) for every nonempty subset A of P∈P P, there exists a P ∈ P such that A ∩ P is a nonempty relatively open subset of A. S If P∈P P = X, then we simply call P an exhaustive partition of X. Given partitions P and Q of a set X, we shall say that P is a refinement of Q if for each P ∈ P there is a Q ∈ Q such that P ⊆ Q. Now, if P and Q are partitions of X, then P ∨ Q := {Y ∈ 2X \ {∅} : Y = P ∩ Q for some P ∈ P and Q ∈ Q} is also a partition of X that is a refinement of both P and Q. Furthermore, if P and Q are exhaustive partitions of a topological space (X, τ), then P ∨ Q is also an exhaustive partition of X. Proposition 3. Every exhaustive partition of a hereditarily Lindelöf space is countable. Proof. Let (X, τ) be a hereditarily Lindelöf topological space and let P be an exhaustive partition of X. Let A be the family of all Q ⊆ P such that Q is a countable partial exhaustive partition of X. Then (A , ⊆) is a nonempty partially ordered set. [3] Fragmentability by the discrete metric 305 Furthermore, from Zorn’s lemma and the fact that (X, τ) is hereditarily Lindelöf, it follows that (A , ⊆) has a maximal element Qmax . S S We claim that Q∈Qmax Q = X (which implies that Qmax = P). Indeed, if Q∈Qmax Q , S X, then X \ ( Q∈Qmax Q) , ∅. Since P is exhaustive, there exists a P ∈ P such that ∅ , S S P ∩ (X \ ( Q∈Qmax Q)) is relatively open in X \ ( Q∈Qmax Q). If we let Q∗ := Qmax ∪ {P}, then Q∗ ∈ A , Qmax ⊆ Q∗ and Qmax , Q∗ . However, this contradicts the maximality of S Qmax . Therefore, Q∈Qmax Q = X and so P = Qmax ∈ A . Theorem 4. Let (X, τ) be a completely regular topological space. Then X is the continuous image of a separable metric space if, and only if, (X, τ) is hereditarily Lindelöf and fragmented by a metric whose topology is at least as strong as τ. Proof. Suppose that X is the continuous image of a separable metric space. Then, clearly, (X, τ) is hereditarily Lindelöf and, by [23, Proposition 2.1], (X, τ) is fragmented by a metric whose topology is at least as strong as τ. Conversely, suppose that (X, τ) is hereditarily Lindelöf and fragmented by a metric d whose topology on X is at least as strong as τ. For each n ∈ N, let Pn be a maximal partial exhaustive partition of X such that d − diam(P) < 1/n for each P ∈ P. Since (X, τ) is fragmented by d, each Pn is in fact an exhaustive partition of X. By passing to a refinement, we may assume that for each n ∈ N, Pn+1 is a refinement of Pn . Furthermore, by Proposition 3, we can write, for each n ∈ N, Pn := {Pkn : k ∈ Ωn }, where ∅ , Ωn ⊆ N. Let Y \ Σ := σ ∈ Ωn : Pnσ(n) , ∅ . n∈N n∈N Endow Σ with the Baire metric d, that is, if σ , σ0 , then d(σ, σ0 ) := 1/n, where n := T min{k ∈ N : σ(k) , σ0 (k)}. Next define f : (Σ, d) → (X, τ) by f (σ) ∈ n∈N Pnσ(n) . Note T σ(n) that f is well defined, since | n∈N Pn | = 1 for all σ ∈ Σ. Clearly, f is a bijection from Σ onto X and, since f (B(σ, 1/n)) ⊆ Pσ(n) (where B(σ, 1/n) := {σ0 ∈ Σ : d(σ, σ0 ) < 1/n)) n σ(n) and d − diam(Pn ) < 1/n, we see that f is continuous on Σ. It is known that fragmentability of a topological space is characterised by the existence of a winning strategy for one of the players (usually called B) in a certain topological game [20, 21]. It is also known that the lack of a winning strategy for the other player (usually called A) in the same game characterises a property that is close to the Namioka property [18, 19]. To be more precise about this, we need the following definition. Let X be a set with two (not necessarily distinct) topologies τ1 and τ2 . On X we will consider the G (X, τ1 , τ2 )-game played between two players A and B. Player A goes first (every time—life is not always fair) and chooses a nonempty subset A1 of X. Player B must then respond by choosing a nonempty relatively τ1 -open subset B1 of A1 . Following this, player A must select another nonempty set A2 ⊆ B1 ⊆ A1 and in turn player B must again respond by selecting a nonempty relatively τ1 -open subset B2 ⊆ A2 ⊆ B1 ⊆ A1 . Continuing this process indefinitely, the players A and B produce 306 W. B. Moors [4] a sequence ((An , Bn ) : n ∈ N) of pairs of nonempty subsets (with Bn relatively τ1 -open in An ) called a play of the G (X, τ1 , τ2 )-game. We shall declare that the player B wins a T T play ((An , Bn ) : n ∈ N) if either (i) n∈N An = ∅ or else (ii) n∈N An = {x} for some x ∈ X and for every τ2 -open neighbourhood U of x there exists an n ∈ N such that An ⊆ U. Otherwise, the player A is said to have won. By a strategy σ for the player B we mean a ‘rule’ that specifies each move of the player B in every possible situation that can occur. Since in general the moves of the player B may depend upon the previous moves of the player A, we shall denote by σ(A1 , A2 , . . . , An ) the nth move of the player B under the strategy σ. We shall call a strategy σ, for the player B, a winning strategy if he/she wins every play of the G (X, τ1 , τ2 )-game, in which they play according to the strategy σ. For a more precise definition of a strategy, see [2]. The main result connecting the G (X, τ1 , τ2 )-game to fragmentability is the following theorem. Theorem 5 [21, Theorem 1.2]. Let τ1 , τ2 be two (not necessarily distinct) topologies on a set X. The space (X, τ1 ) is fragmentable by a metric whose topology is at least as strong as τ2 if, and only if, the player B has a winning strategy in the G (X, τ1 , τ2 )-game played on X. Throughout the remainder of this paper we will be interested in the case when τ2 is the discrete topology—which we will denote by τd . We have seen in Theorem 1 that fragmentability by a metric that generates the discrete topology (or, equivalently, the existence of a winning strategy for the player B in the G (X, τ1 , τd )-game) reduces to the study of σ-scattered spaces. However, it might be interesting to see whether the lack of a winning strategy for the player A in the G (X, τ1 , τd )-game leads to anything more interesting. Our next result requires two more auxiliary notions. The first is the notion of quasicontinuity. Suppose that f : (X, τ) → (Y, τ0 ) is a function acting between topological spaces (X, τ) and (Y, τ0 ). Then we say that f is quasi-continuous if for each open set W in Y, f −1 (W) ⊆ int( f −1 (W)) [16]. The second notion that is needed is that of an α-favourable space, whose precise definition can be found in [19]. Theorem 6 [19, Theorem 1]. Let (X, τ) be a Hausdorff regular space. Then the following are equivalent: the G(X, τ, τd )-game is A-unfavourable; for every quasi-continuous mapping f : Z → (X, τ) from a complete metric space Z there is a nonempty open subset U such that f is constant on U; (iii) for every quasi-continuous mapping f : Z → (X, τ) from an α-favourable space Z there is a nonempty open subset U such that f is constant on U; (iv) for every continuous mapping f : Z → (X, τ) from an α-favourable space Z there is a nonempty open subset U such that f is constant on U. (i) (ii) If the topology τ is metrisable, then we have the following theorem. [5] Fragmentability by the discrete metric 307 Theorem 7. Let (X, τ) be a metrisable space. Then the following are equivalent: the G(X, τ, τd )-game is A-unfavourable; for every continuous mapping f : Z → (X, τ) from a complete metric space Z there is a nonempty open subset U such that f is constant on U; (iii) for every quasi-continuous mapping f : Z → (X, τ) from a complete metric space Z there is a nonempty open subset U such that f is constant on U; (iv) for every quasi-continuous mapping f : Z → (X, τ) from an α-favourable space Z there is a nonempty open subset U such that f is constant on U; (v) for every continuous mapping f : Z → (X, τ) from an α-favourable space Z there is a nonempty open subset U such that f is constant on U. (i) (ii) Proof. Clearly, (iii) ⇒ (ii) and so by Theorem 6 it is sufficient to show that (ii) ⇒ (iii). Suppose that Z is a complete metric space and f : Z → (X, τ) is quasi-continuous. Since (X, τ) is metrisable, we have from [1] that there exists a dense Gδ subset G of Z such that f is continuous at each point of G. Now, by [15, page 208] or [34, page 164], there exists a complete metric d on G that generates the relative topology on G. Next, by our assumption, f |G : G → X has a nonempty open subset U of G such that f |G (U) =: {x} is a singleton. Let U ∗ be any open subset of Z such that U ∗ ∩ G = U. Since f is quasi-continuous on Z, it follows (see for example [28]) that f (U ∗ ) ⊆ f |G (U) = {x} = {x}. Hence, f is constant on U ∗ , which completes the proof. We may now apply this result along with the definition of a perfect set to obtain the following useful characterisation. Recall that a subset of a topological space (X, τ) is called perfect if it is closed and does not have any isolated points. Corollary 8. Let (X, τ) be a metrisable space. Then the G(X, τ, τd )-game is Aunfavourable if, and only if, X does not contain any perfect compact subsets. Proof. Suppose that the G(X, τ, τd )-game is A-unfavourable. In order to obtain a contradiction, let us suppose that X contains a perfect compact set Z. We shall consider the identity mapping f : Z → Z defined by f (z) := z for all z ∈ Z. Now, since Z is a perfect set, it does not have any isolated points and so we have a continuous nowhereconstant function defined on a complete metric space. This contradicts part (ii) of Theorem 7. For the converse, let us start by assuming that X does not contain any perfect compact subsets. From Theorem 7, it is sufficient to show that for any complete metric space M and any continuous function f : M → X there is a nonempty open subset U of M such that f is constant on U. Let (M, ρ) be a complete metric space. In order to obtain a contradiction, let us suppose that f : M → X is not constant on any nonempty open subset of M. Let D be the set of all finite sequences of zeros and ones. We shall inductively (on the length |d| of d ∈ D) define a family {Cd : d ∈ D} of nonempty open subsets of M such that: 308 W. B. Moors [6] (i) ρ − diam(Cd ) < 1/2|d| ; (ii) ∅ , Cd0 ∩ Cd1 ⊆ Cd0 ∪ Cd1 ⊆ Cd ; (iii) f (Cd0 ) ∩ f (Cd1 ) = ∅. Base step: let C∅ be a nonempty open subset of M with ρ − diam(C∅ ) < 1/20 , where the sequence of length zero is denoted by ∅. Assuming that we have already defined the nonempty open sets Cd satisfying (i), (ii) and (iii) for all d ∈ D with |d| ≤ n, we proceed to the inductive step. Inductive step: fix d ∈ D of length n. Therefore, there exist points c0 and c1 in Cd such that f (c0 ) , f (c1 ). From the continuity of f , we can choose open neighbourhoods Cd0 of c0 and Cd1 of c1 such that conditions (i), (ii) and (iii) are satisfied. This completes the induction. T S Now, for each n ∈ N, let Kn := {Cd : d ∈ D and |d| = n} and K := n∈N Kn . Then K is closed and totally bounded and hence compact. Furthermore, K is perfect and f is one-to-one on K. Therefore, f (K) is a perfect compact subset of X, which contradicts our assumption concerning X. Therefore, f must be constant on some nonempty open subset U of M. In order to state our last result, we need to recall the definition of a Bernstein set. A subset B of R is called a Bernstein set if neither B nor its complement contains a perfect compact subset [32, page 23]. In [32], the construction of a Bernstein set is given. It is also easy to check that every Bernstein set is uncountable. Corollary 9. Let B be a Bernstein subset of R endowed with the relative topology τ inherited from R with its usual topology. Then neither player (A nor B) has a winning strategy in the G(B, τ, τd )-game played on B. 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